Partial direct product difference sets and almost quaternary sequences

Abstract

In this paper, we study the \begin{document}$ m $\end{document} -ary sequences with (non-consecutive) two zero-symbols and at most two distinct autocorrelation coefficients, which are known as almost \begin{document}$ m $\end{document} -ary nearly perfect sequences. We show that these sequences are equivalent to \begin{document}$ \ell $\end{document} -partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zero-symbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for \begin{document}$ \ell $\end{document} -PDPDS. Finally, we present two construction methods for a family of almost quaternary sequences with at most two out-of-phase autocorrelation coefficients.